Construction of Butson matrices using Fourier matrices as input
Farouk Adda
TL;DR
This work tackles the construction of larger-order Butson matrices $BH(m,n)$ by proposing two Scarpis-inspired schemes that use existing matrices of order $n$ to produce matrices of orders $n(n-1)$ and $n(\frac{n}{2}-1)$ when $m$ and $n$ are even. The first construction, via $\Phi$, builds $BH(m,(n-1)n)$ from a given $BH(m,n)$ using a complete tensor set of Latin squares eligible for Scarpis construction (LSESC) and a Kronecker-based block assembly, with a one-input and a two-input variant. The second construction, via $\Psi$, yields $BH(m,((n/2)-1)n)$ from inputs satisfying certain C1/C2-type conditions (including Fourier matrices for suitable $n$) and employs a layered block structure with matrices $E_i$ and $T_{x_d,x_{d+1}}$, along with a two-input extension. The results connect Latin-square combinatorics (LSESC and MOLS) with complex Hadamard-type matrices, and yield Hadamard-matrix corollaries and enumeration insights for the produced families. Overall, the paper broadens the repertoire of known BH matrices and provides concrete, input-driven pathways to generate higher-order complex Hadamard matrices from foundational Fourier/Hadamard inputs.
Abstract
Butson matrices are square orthogonal matrices, denoted by $BH(m,n)$, whose entries are the complex $m$th roots of unity and satisfy the condition\\ $BH(m,n)\cdot{BH(m,n)}^*=nI_n$, where ${BH(m,n)}^*$ is the conjugate transpose of $BH(m,n)$ and $I_n$ is the identity matrix. In this work, we propose constructions for $BH(m,(n-1)n)$ then $BH(m,(\frac{n}{2}-1)n)$, when $n$ and $m$ are even numbers, using the existing $BH(m,n)$. For each case, we provide two construction methods: one uses a single input Butson matrix, and another uses two input Butson matrices. Moreover, we present some results about the construction of Hadamard matrices.